Variance Futures listed on Safex: a short note
V1.0
Dr Antonie Kotzé1
and
Angelo Joseph
September-2008
1. Introduction
2. Volatility as an Investable Asset Class
3. The Variance Swap Mechanism
4. Variance Swap Pricing in Theory
5. Pricing in Practice
6. Mark to Market
7. Initial Margin Requirements
8. Profit and Loss
9. Trading a Standard Variance Future
10. References
Annexure A: Example of a Variance Future Term Sheet
1
Consultants from Financial Chaos Theory (www.quantonline.co.za) to the JSE
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4
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1. Introduction
Volatility is a measure of the risk or uncertainty and it has an important role in the
financial markets. Volatility is defined as the variation of an asset's returns – it
indicates the range of a return's movement. Large values of volatility mean that
returns fluctuate in a wide range – in statistical terms, the standard deviation is such
a measure and offers an indication of the dispersion or spread of the data2.
Volatility forms an integral part of the Black-Scholes-Merton option pricing model [BS
73]. Even though the model treats the volatility as a constant over the life of the
contract, these three pioneers knew that volatility changes over time. As far back as
1976, Fischer Black wrote [Bl 76]
“Suppose we use the standard deviation … of possible future returns on a stock …
as a measure of volatility. Is it reasonable to take that volatility as constant over time?
I think not."
Black&Scholes defined volatility as the standard deviation because it measures the
variability in the returns of the underlying asset [BS 72]. They determined the
historical volatility and used that as a proxy for the expected or implied volatility in
the future. Since then the study of implied volatility has become a central
preoccupation for both academics and practitioners [Ga 06].
Figure 1 shows a plot of the 3 month historical volatility for the JSE/FTSE Top 40
index since January 2007 using daily data. It is clear that volatility is not constant.
35%
30%
Volatility (%)
25%
20%
15%
Sep-08
Aug-08
Jul-08
Jun-08
May-08
Apr-08
Mar-08
Feb-08
Jan-08
Dec-07
Nov-07
Oct-07
Sep-07
Aug-07
Jul-07
Jun-07
May-07
Apr-07
Mar-07
Feb-07
Jan-07
10%
Date
Figure 1: Three month historical rolling volatility for the JSE/FTSE Top 40 index
from January 2007 to September 2008.
2
The standard deviation is one of the fundamental elements of the Gauss or normal distribution curve – it
describes the width of the famous bell curve.
Volatility is, however, statistically persistent, i.e., if it is volatile today, it should
continue to be volatile tomorrow. This is also known as volatility clustering and can
be seen in Figure 2. This is a plot of the logarithmic returns of the Top 40 index since
June 1995 using daily data.
0.1
0.05
Return
0
-0.05
-0.1
-0.15
Figure 2: JSE/FTSE Top 40 daily log returns from June 1995 to September 2008.
Note the 13.3% negative return on 28 October 1997 – start of the Asian crises.
But, what is this ``volatility"? As a concept, volatility seems to be simple and intuitive.
Even so, volatility is both the boon and bane of all traders --- you can't live with it and
you can't really trade without it. Without volatility, no trader can make money! Most of
us usually think of ``choppy" markets and wide price swings when the topic of
volatility arises. These basic concepts are accurate, but they also lack nuance.
How do we define the volatility (standard deviation) as being good, or acceptable, or
normal? By many standards, a large standard deviation indicates a non-desirable
dispersion of the data, or a wide (wild) spread. It is said that such a phenomenon is
very volatile – such phenomena are much harder to analyze, or define, or control.
Volatility also has many subtleties that make it challenging to analyze and implement
[Ne 97]. The following questions immediately come to mind: is volatility a simple
intuitive concept or is it complex in nature, what causes volatility, how do we
estimate volatility and can it be managed?
The management of volatility is currently a topical issue. Due to this, many
institutional and individual investors have shown an increased interest in volatility as
an investment vehicle.
2. Volatility as an Investable Asset Class
Apart from its prominent role as financial risk measure, volatility has now become
established as an asset class of its own. Initially, the main motive for trading volatility
was to manage the risk in option positions and to control the Vega exposure
independently of the position’s delta and gamma. With the growth of the volatility
trading segment, other market participants became aware of volatility as an
investment vehicle [HW 07]. The reason is that volatility has peculiar dynamics:
-
It increases when uncertainty increases.
Volatility is mean reverting - high volatilities eventually decrease and low ones
will likely rise to some long term mean.
Volatility is often negatively correlated to the stock or index level.
Volatility clusters.
How can investors get exposure to volatility? Traditionally, this could only be done
by taking positions in options, and then, by delta hedging the market risk. Delta
hedging, however, is at best, inaccurate due to the Black&Scholes assumptions like
continuous trading. Taking a position in options, therefore, provides a volatility
exposure (Vega risk) that is contaminated with the direction of the underlying stock
or index level.
Variance swaps emerged as a means of obtaining a purer volatility exposure. These
instruments took off as a product in the aftermath of the Long Term Capital
Management (LTCM) meltdown in late 1998 during the Asian crises. Some
additional features are
•
•
•
variance is the square of volatility;
simple payoffs; and
simple replication via a portfolio of vanilla options.
These are described in the next few sections.
3. The Variance Swap Mechanism
Variance swaps or variance futures (also called variance contracts) are equity
derivative instruments offering pure exposure to daily realised future variance.
Variance is the square of volatility (usually denoted by the Greek symbol σ 2 ). At
expiration, the swap buyer receives a payoff equal to the difference between the
annualised variance of logarithmic stock returns and the swap rate fixed at the
outset of the contract. The swap rate (or delivery price) can be seen as the fixed leg
of the swap and is chosen such that the contract has zero present value.
In short, a variance swap is not really a swap at all but a forward contract on realised
annualised variance. A long position’s payoff at expiration is equal to [DD 99]
[
VNA σ R2 − K var
]
(1)
where VNA is the Variance Notional Amount, σ R2 is the annualised non-centered
realised variance of the daily logarithmic returns on the index level and K var is the
delivery price. Note that VNA is the notional amount of the swap in Rand per
annualised variance point. The holder of a variance swap at expiration receives VNA
Rands for every point by which the stock’s realised variance has exceeded the
variance delivery price.
A capped variance swap is one where the realised variance is capped at a
predefined level. From Eq. (1) we then have
[
]
VNA min (cap, σ R2 ) − K var .
The realised variance is defined by
252
σ =
n
2
R
  S 2 
ln i  
∑
i =1   S i −1  


n
(2)
with S i the index level and n the number of data points used to calculate the
variance [BS 05].
If we scrutinised Eq. (2), we see that the mean logarithmic return is dropped if we
compare this equation with the mathematically correct equation for variance. Why?
Firstly, its impact on the realised variance is negligible. Secondly, this omission has
the benefit of making the payoff perfectly additive3. Another reason for ditching the
mean return is that it makes the estimation of variance closer to what would affect a
trader's profit and loss. The fourth reason is that zero-mean volatilities/variances are
better at forecasting future volatilities. Lastly Figlewski argues that, since volatility is
measured in terms of deviations from the mean return, an inaccurate estimate of the
mean will reduce the accuracy of the volatility calculation [Fi 94]. This is especially
true for short time series like 1 to 3 months (which are the time frames used by most
traders to estimate volatilities).
Note, historical volatility is usually taken as the standard deviation whilst above we
talk about the variance. The question is: why is standard deviation rather than
variance often a more useful measure of variability? While the variance (which is the
square of the standard deviation) is mathematically the "more natural" measure of
deviation, many people have a better "gut" feel for the standard deviation because it
has the same dimensional units as the measurements being analyzed. Variance is
interesting to scientists, because it has useful mathematical properties (not offered
by standard deviation), such as perfect additivity (crucial in variance swap
instrument development). However, volatility is directly proportional to variance.
3
Suppose we have the following return series: 1%, 1%, -1%, -1%. Using variance with the sample mean, the
variance over the first two observations is 0, and the variance over the last two observations is zero. However
the total variance with sample mean over all 4 periods is 0.01333%, which is clearly not zero. If we assumed the
sample mean to be zero, we get perfect variance additivity.
4. Variance Swap Pricing in Theory
The price of the variance swap per variance notional at the start of the contract is the
delivery variance K var . So how do we find the delivery price such that the swap is
immune to the underlying index level? Carr and Madan came up with a static hedge
by considering the following ingenious argument [CM 98]: We know that the
sensitivity of an option to volatility, Vega, is centered (like the Gaussian bell curve)
around the strike price and will thus change daily according to changes in the stock
or index level. We also know that the higher the strike, the larger the Vega. If we can
create a portfolio of options with a constant Vega, we will be immune to changes in
the stock or index level – a static hedge. Further, by induction, it turns out that the
Vega is constant for a portfolio of options inversely weighted by the square of their
strikes. This hedge is independent of the stock level and time.
From the previous argument we deduce that the fair price of a variance swap4 is the
value of a static option portfolio, including long positions in out-the-money (OTM)
options, for all strikes from 0 to infinity. The weight of every option in this portfolio is
the inverse square of its strike. Demeterfi and Derman et. al. discretised Carr and
Madan’s solution and set out all the relevant formulas and this will be the subject of
the technical note [DD 99].
5. Pricing in Practice
Here are some practical hints to consider when estimating the implied variance, K var ,
through the portfolio of options:
1. The chosen strikes have fix increments that are not infinitesimally small. This
introduces a discretisation error5 in the approximation of the fair variance. The
smaller the increments the smaller the error. Safex will use an increment of
10 index points in the valuation of variance futures on indices.
2. A strike range is chosen and is thus also finite. This introduces a truncation
error. This range is heavily dependent on the range of volatilities available.
Note that the fair variance accuracy is more sensitive to the strike range, than
the strike increments. In other words, the accuracy of the fair variance is more
prone to truncation errors.
Safex publishes volatility skews for a strike range of 70% to of 130%. The
strike range used will thus be from 70% moneyness to 130% moneyness.
This will ensure that Vega is constant over this range of strikes. We must be
aware that even though truncation is the main error, fixing Vega to the strike
range combined with optimal strike spacing, ensures that Vega is constant
over that strike range.
3. For a small number of symmetrical options (< 20) we have that the lower the
put bound strike the more over-estimated the fair variance. This overestimation is less pronounced the higher the number of symmetrical options.
4
The fair price of a variance swap, given that it is the price of variance in the future, it is also referred to as the
fair value of future variance.
5
We implicitly use the term error, here and not uncertainty. This is so because an error implies that the
theoretical true value is known, whereas uncertainty does not.
4. For a low put bound strike (< 40%) we have that the higher the number of
symmetrical options, the more underestimated the fair variance. This underestimation is considerably less pronounced the higher the left wing bound.
Finding the optimal strike increments (discretisation) and the strike range (minimise
truncation errors) are a trade-off between over and under estimating the fair variance.
6. Mark to Market
Any time, t , during the life of a variance swap, expiring at time T , the price of the
variance swap, can be decomposed into a realised variance part (the already
crystallised variance), and an implied variance part (the variance part that must still
be crystallised or future/fair variance). Because variance is additive the price of the
variance swap of 1 variance notional at time t (or the price of a new swap maturing
at time (T-t), is
Vmtm =
t 2 T −t
σR +
K var (t , T )
T
T
(3)
100%
0%
90%
10%
80%
20%
Realised
70%
30%
60%
40%
50%
50%
40%
60%
30%
70%
Implied
20%
80%
10%
90%
0%
0%
At Swap Inception
20%
40%
60%
Proportion Time
80%
Proportion Realised Variance
Proportion Implied Variance
At contract initiation the value will be derived from 100% implied variance, as the
contract moves closer to expiry realised variance will play an increasingly important
role. This is shown in Figure 3 below. Variation margin is thus the difference
between the value of the variance future today and what it was yesterday as given
by Eq. (3).
100%
100%
At Swap Expiry
Figure 3: As time passes, realised variance start to play the dominant role in the
value of a variance future.
7. Initial Margin Requirements
The initial margin requirement is determined in using the underlying’s historical data
to determine a volatility of volatility like parameter, the Risk Parameter λ . The Risk
Parameter is currently set at 10%. The initial margin requirement ( IMR ) per contract
for a variance future is then given by
IMR = λ ∗ K market ∗ VPV .
(4)
Here, K market is the market value of implied variance and VPV is the Variance Point
Value. Note, the VPV is a quantity similar to the current R10 per point used for index
futures. It is currently set at VPV = R1 for the Safex variance futures.
Most traders trading a variance future usually start with a Vega Rand amount they
want to hedge – this might be due to a book of options. We can then define [Wi 08]
VNA =
VA
2 K var
= C ∗ VPV
(5)
with VA the Vega amount to be hedged in Rand. Here, K var is in absolute terms e.g.,
if the volatility is 30% then K var = 30 2 = 900 . From this
C=
VA
VNA
=
.
VPV 2 K var VPV
(6)
Eq. (6) is used to determine the number of contracts, C , to trade.
From Eq. (3) and Figure 3 we deduce that the risk in a variance future dwindles as
we approach the expiry date – there is more certainty in the outcome because of the
realized variance part. For 3 months and 6 months variance future the initial margins
are then reduced in accordance with the tables in Table 1 and Table 2 respectively.
Month
Month
Month
0 to 1
1 to 2
2 to 3
Margin as % of IMR
100%
75%
50%
Table 1: Reduction in initial margin requirements for a 3 month variance future.
Month
Month
Month
Month
Month
Month
0 to 1
1 to 2
2 to 3
3 to 4
4 to 5
5 to 6
Margin as % of IMR
100%
90%
80%
70%
60%
50%
Table 2: Reduction in initial margin requirements for a 6 month variance future.
8. Profit and Loss
To determine the profit and loss for a variance future, we return to Eq. (1) and (3).
We then have
PL = VNA [Vmtm − K ] = C ∗ VPV [Vmtm − K ]
(7)
where Vmtm is the daily mark-to-market variance level as calculated using Eq. (3) and
set by Safex on a daily basis. K is the delivery price or strike price set at initiation of
the contract.
9. Trading a Standard Variance Future
Eq. (6) determines the number of contracts to trade if a certain Rand amount of
Vega exposure needs to be hedged. This holds at the initiation of the variance
futures contract. Also, remember that at initiation the value of the variance future is
zero. The delivery price, is then found from Eq. (1).
An investor buying into an existing variance future (after initiation) will buy an
instrument that already has value due to the realised variance part in Eq. (2). His
future’s value needs to be zero as well but the delivery price is fixed. The only way to
get a value of zero is to change the number of contracts to trade. The number of
contracts is then given by
C new =
VA
2 ∗ VPV ∗
Vmtm
(8)
K
where Vmtm is the daily mark-to-market variance level as calculated through Eq. (3)
and set by Safex on a daily basis. K is the delivery price or strike price set when the
existing variance future was initiated. VA is the Vega Rand amount to be hedged.
10. References
[BS 72] Fischer Black and Myron Sholes, The Valuation of Option Contracts and a
test of Market Efficiency, Proceedings of the Thirtieth Annual Meeting of he
American Finance Association, 27-29 Dec. 1971, J. of Finance, 27, 399 (1972)
[BS 73] Fischer Black and Myron Sholes, The Pricing of Options and Corporate
Liabilities, J. Pol. Econ., 81, 637 (1973)
[BS 05] Sebastien Bossu, Eva Strasser and Regis Guichard, Just What You Need
To Know About Variance Swaps, JPMorgan London, Equity Derivatives, February
(2005)
[Bl 76] Fischer Black, The Pricing of Commodity Contracts, J. Fin. Econ., 3, 167
(1976)
[CM 98] Peter Carr and Dilip Madan, Towards a Theory of Volatility Trading,
in Volatility: New Estimation Techniques for Pricing Derivatives,
edited by R. Jarrow, 417-427 (1998).
[DD 99] Kresimir Demeterfi, Emanuel Derman, Michael Kamal and Joseph Zou,
More Than You Ever Wanted To Know About Volatility Swaps, Quantitative
Strategies Research Notes, Goldman Sachs & Co, March, (1999).
[DK 96] Emanuel Derman, Michael Kamal, I. Kani and Joseph Zou, Valuing
Contracts with Payoffs Based on Realised Volatility, Global Derivatives, Quarterly
Review, Equity Derivatives Research, Goldman Sachs & Co, July (1996)
[Fi 94] S. Figlewski, Forecasting Volatility using Historical Data, Working Paper,
Stern School of Business, New York University, (1994)
[Ga 06] Jim Gatheral, The Volatility Surface: a practitioner’s guide, Wiley Finance
(2006)
[HW 07] Reinhold Hafner and Martin Wallmeier, Optimal Investments in Financial
Instruments with Heavily Skewed Return Distributions: The Case of Variance Swaps,
Working Paper, Risklab Germany GmbH (June 2007)
[Mo 05] Nicolas Mougeot, Volatility Investing Handbook, BNP PARIBAS, Equity and
Derivatives Research, September, (2005).
[Ne 97] Izrael Nelken, Ed. Volatility in the Capital Markets, Glenlake Publishing
Company (1997)
[Wi 08] Wikipedia, the free encyclopedia, http://en.wikipedia.org/wiki/Variance_swap
Annexure A: Example of a Variance Future Term Sheet
JSE EQUITY DERIVATIVES MARKET NOTICE
Number
Date
24 July 2008
New Variance Futures
The following new future has been added to the list with immediate effect and will be
available for trading on Friday 25 July 2008:
Summary Contract Specifications
Futures Contract
Code
XXXX
Underlying
Instrument
FTSE/JSE TOP40 Index future
Variance Point
Value, VPV
1 Variance point = 1 Rand
Trade Date
14 August 2008
Expiry Dates &
Times
17:00 on 18 September 2008
Quotations
Variance points to two decimals.
Minimum Price
Movement
R1 (0.01 in variance points)
Expiry Valuation
Method
Realised variance (RV) as calculated by the JSE
Limited over the contract period (Formula
Appendix A)
Final Equity
Payment on Expiry
Clearing House
Fees
Initial Margin per
Contract
Class Spread
Margin
C * VPV * [RV – K]
TBA
R90
R
V.S.R
3.5
Volatility Strike, VK
30
Volatility Cap, VC
NA
Variance Strike, K
(K = VK2)
900
Variance Cap, KC
NA
Number of trading
observation days
Communication
Code
All trading days between and inclusive of the trade
and expiration dates
Appendix
Realised variance will be calculated as follows:
252 n−1   Si+1 
RV = 10000×
∑ ln 
n − 1 i=1   Si 

2



Where:
i = each observation date
n = number of observation dates as of trade date
S i = the closing level of the underlying index
future on the i -th observation date
S n = the closing level on the exchange of
the index future on the expiration date